Calculating the efficiency and productivity of decision making units with negative data

Although non-negative data are fundamentally indispensable for determining the Malmquist productivity index (MPI), the observed values are sometimes negative in the real-world problems. In this paper, we reformulate the conventional Malmquist productivity index in data envelopment analysis (DEA) problem with negative data. So, first we want to introduce a non-radial efficiency model with negative data, then we use it in the Malmquist productivity index. At the end, we have tested the new proposed approach by the case study and applied to the productivity analysis of the 28 cement companies where are located in Iran’s Burs evolved between 2012 and 2013, because some of these companies have one negative output. In the analysis of the case study, we show that the index Malmquist (productivity) of companies is measured correctly. an‎d because the index Malmquist calculations are done by computing efficiency, therefore it can be resulted that the efficiency of companies with negative data is measured correctly.